10.3One-Way Analysis of Variance 445
will be a chi-square random variable withm−1 degrees of freedom. That is, if we define
SSbby
SSb=n
∑m
i= 1
(Xi.−X..)^2
then it follows that
when H 0 is true,
SSb/σ^2 is chi-square withm−1 degrees of freedom
From the above we obtain that whenH 0 is true,
E[SSb]/σ^2 =m− 1
or, equivalently,
E[SSb/(m−1)]=σ^2 (10.3.4)
So, whenH 0 is true,SSb/(m−1) is also an estimator ofσ^2.
Definition
The statistic
SSb=n
∑m
i= 1
(Xi.−X..)^2
is called thebetween samples sum of squares. WhenH 0 is true,SSb/(m−1) is an estimator
ofσ^2.
Thus we have shown that
SSW/(nm−m) always estimatesσ^2
SSb/(m−1) estimatesσ^2 whenH 0 is true
Because* it can be shown thatSSb/(m−1) will tend to exceedσ^2 whenH 0 is not true, it
is reasonable to let the test statistic be given by
TS=
SSb/(m−1)
SSW/(nm−m)
and to rejectH 0 whenTSis sufficiently large.
* A proof is given at the end of this section.